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In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, modified simple equation method has been implemented, to find the exact solutions of these equations, in the sense of modified Riemann-Liouville derivative. For applications, the exact solutions of time-space fractional derivative Burgers’ equation and time-space fractional derivative foam drainage equation have been discussed. Moreover, it can also be concluded that the proposed method is easy, direct and concise as compared to other existing methods.

Nonlinear partial differential equations have shown a variety of applications in almost every field of life, such as in electromagnetics, acoustics, electrochemistry, cosmology, biological and material science [

Knowing the importance of differential equations of fractional order, lots of authors are working to find the exact or numerical solutions of the equations. For examples, the adomian decomposition method [

In this article, a new approach has been developed to find the exact solutions of nonlinear partial differential equations of fractional order by the fractional complex transformation [

and time-spce fractioanl derivative foam drainage equation [

The rest of the article is organized as follows, in section 2 the basic definitions and properties of the fractional theory are considered regrading to modified Riemann-Liouville derivative. In section 3, the modified simple equation method has been proposed to find the exact solutions for NPDEs of fractional order with the help of fractional complex transformation. The two applications are being considered to find the exact solution in section 4. In last section 5, the conclusion has been drawn.

In this section, the extended method has been applied in the sense of the Jumarie’s modified Riemann-Liouville derivative of order

Definition 2.1 A real function

Definition 2.2 The Jumarie’s modified Riemann-Liouville derivative, of order

Moreover, some properties for the modified Riemann-Liouville derivative have also been given as follows:

In this section, the modified simple equation method [

For this, we consider the following NPDE of fractional order:

where u is an unknown function and P is a polynomial of u and its partial fractional derivatives along with the involvement of higher order derivatives and nonlinear terms.

To find the exact solutions, the method can be performed using the following steps.

Step 1: First, we convert the NPDE of fractional order into nonlinear ordinary differential equations using fractional complex transformation introduced by Li et al. [

The travelling wave variable

where K, L and M are non-zero arbitrary constants, permits us to reduce Equation (3.2) to an ODE of

Step 2: Suppose that the solution of Equation (3.3) can be expressed as a polynomial of

where

Step 3: The homogeneous balance can be used, to determine the positive integer m, between the highest order derivatives and the nonlinear terms appearing in (3.4).

Step 4: After the substitution of (3.4) into (3.3), we collect all the terms with the same order of

Step 5: After solving the system of algebraic equations, the variety of exact solutions can be celebrated.

In the following subsections, two applications (given in Equations (1.1) and (1.2)) are being considered to find the exact solutions by the proposed method.

In this section, the modified simple equation method has been applied to construct the exact solutions for the nonlinear space-time fractional Burgers’ Equation (1.1). It can be observed that the fractional complex transform

where K and L are constants, permits to reduce the Equation (1.1) into an ODE of the following form:

Now by calculating the homogeneous balance (i.e.,

where

and

The above Equation (4.6), yields the value

The general solution of the Equation (4.4) is

While

For, the value

which also gives the same results.

Applying the fractional complex transformation on the Equation (1.2), which reduces into the following form:

Now by calculating the homogeneous balance, which is

where

and

The above Equation (4.13), yields the value

Case 1: The general solution of the Equation (4.10) is

where

Case 2: For the value

while

Case 3: For the value

where

Which are the required results.

The modified simple equation method has been extended to solve the nonlinear partial differential equation of fractional order, in the sense of modified Riemann-Liouville derivative. First, the fractional complex transformation has been used to convert the fractional order differential equations into ordinary differential equations. Then, the modified simple equation method has been used to find the exact solutions. The two applications have been considered to find the new exact solutions for the nonlinear time-space fractional derivative Burgers’ equation and time-space fractional derivative foam drainage equation. It can also be concluded that the proposed method is very simple, reliable and a variety of exact solutions to NPDEs of fractional order are proposed.